Abstract
We prove joint large deviation principle for the empirical pair measure and empirical locality measure of the near intermediate coloured random geometric graph models on n points picked uniformly in a d-dimensional torus of a unit circumference. From this result we obtain large deviation principles for the number of edges per vertex, the degree distribution and the proportion of isolated vertices for the near intermediate random geometric graph models.
Highlights
In this article we study the coloured geometric random graph CGRG, where n points or vertices or nodes are picked uniformly at random in [0, 1]d, colours or spins are assigned independently from a finite alphabet Σ and any two points with colours a1, a2 ∈ Σ distance at most rn(a1, a2) apart are connected
Doku-Amponsah (2015) proved joint large deviation principle for empirical pair measure and the empirical locality measure of the CGRG, where n points are uniformly chosen in [0, 1]d, colours or spins are assigned by drawing without replacement from the pool of, say, nνn(a1) colours, and nωn(a1, a2) edges, a1, a2 ∈ Σ, are randomly inserted among the points for some colour law νn : Σ → [0, 1] and edge law ωn : Σ × Σ → [0, ∞)
We have proved joint large deviation principle for the empirical pair measure and empirical locality measure of the near intermediate CGRG models
Summary
In this article we study the coloured geometric random graph CGRG, where n points or vertices or nodes are picked uniformly at random in [0, 1]d, colours or spins are assigned independently from a finite alphabet Σ and any two points with colours a1, a2 ∈ Σ distance at most rn(a1, a2) apart are connected. Refer to (Doku-Amponsah and Moerters) for similar result for the coloured random graphs. From this large deviation results we obtain LDPs for graph quantities. We use (Biggins, 2004, Theorem 5(b)) to mix Theorem 2.4 and the result (Doku-Amponsah, 2015, Theorem 2.1) to obtain the full joint LDP for empirical pair measure and the empirical locality measure of CGRG model.
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